Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, B

Average Error: 8.0 → 0.0

Time: 4.0s

Precision: binary64

Cost: 712

Math

\[\frac{x \cdot y}{y + 1} \]

↓

\[\begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -53455814794.271835:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 44690398.39284847:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* x y) (+ y 1.0)))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- x (/ x y))))
   (if (<= y -53455814794.271835)
     t_0
     (if (<= y 44690398.39284847) (* y (/ x (+ y 1.0))) t_0))))

C

double code(double x, double y) {
	return (x * y) / (y + 1.0);
}

↓

double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -53455814794.271835) {
		tmp = t_0;
	} else if (y <= 44690398.39284847) {
		tmp = y * (x / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (y + 1.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (x / y)
    if (y <= (-53455814794.271835d0)) then
        tmp = t_0
    else if (y <= 44690398.39284847d0) then
        tmp = y * (x / (y + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return (x * y) / (y + 1.0);
}

↓

public static double code(double x, double y) {
	double t_0 = x - (x / y);
	double tmp;
	if (y <= -53455814794.271835) {
		tmp = t_0;
	} else if (y <= 44690398.39284847) {
		tmp = y * (x / (y + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	return (x * y) / (y + 1.0)

↓

def code(x, y):
	t_0 = x - (x / y)
	tmp = 0
	if y <= -53455814794.271835:
		tmp = t_0
	elif y <= 44690398.39284847:
		tmp = y * (x / (y + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(y + 1.0))
end

↓

function code(x, y)
	t_0 = Float64(x - Float64(x / y))
	tmp = 0.0
	if (y <= -53455814794.271835)
		tmp = t_0;
	elseif (y <= 44690398.39284847)
		tmp = Float64(y * Float64(x / Float64(y + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (y + 1.0);
end

↓

function tmp_2 = code(x, y)
	t_0 = x - (x / y);
	tmp = 0.0;
	if (y <= -53455814794.271835)
		tmp = t_0;
	elseif (y <= 44690398.39284847)
		tmp = y * (x / (y + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(x - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -53455814794.271835], t$95$0, If[LessEqual[y, 44690398.39284847], N[(y * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	8.0
Target	0.0
Herbie	0.0

\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < -53455814794.271835 or 44690398.3928484693 < y

   1. Initial program 16.6

      \[\frac{x \cdot y}{y + 1} \]

   2. Taylor expanded in y around inf 0.0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{y} + x} \]

   3. Simplified 0.0

      \[\leadsto \color{blue}{x - \frac{x}{y}} \]
      Proof
      (-.f64 x (/.f64 x y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 x (neg.f64 (/.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 x y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 x y)) x)): 0 points increase in error, 0 points decrease in error

   if -53455814794.271835 < y < 44690398.3928484693

   1. Initial program 0.0

      \[\frac{x \cdot y}{y + 1} \]

   2. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\frac{x}{y + 1} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -53455814794.271835:\\ \;\;\;\;x - \frac{x}{y}\\ \mathbf{elif}\;y \leq 44690398.39284847:\\ \;\;\;\;y \cdot \frac{x}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	712

\[\begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -48.789773808953534:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.0120801136950827 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \left(x - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.1
Cost	584

\[\begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -48.789773808953534:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.0120801136950827 \cdot 10^{-5}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -48.789773808953534:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.0120801136950827 \cdot 10^{-5}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	0.0
Cost	448

\[x \cdot \frac{y}{y + 1} \]

Alternative 5
Error	0.1
Cost	448

\[\frac{x}{\frac{y + 1}{y}} \]

Alternative 6
Error	31.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022313 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) (- (/ x (* y y)) (- (/ x y) x))))

  (/ (* x y) (+ y 1.0)))